23Models - Acollectionofpotentiallyusefulmodels

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Unformatted text preview: Acollectionofpotentiallyusefulmodels Wevealreadyseentwoverycommonmixedmodels: forsubsampling fordesignedexperimentswithmultipleexperimentalunits Herearethreemoregeneralclassesofmodels Randomcoefficientmodels,akamulti-levelmodels Modelsforrepeatedexperiments Modelsforrepeatedmeasures Forcomplicatedproblems,mayneedtocombineideas c 2011Dept.Statistics(IowaStateUniversity) Stat511section23 1/26 Randomcoefficientmodels Aregressionwhereallcoefficientsvarybetweengroups Example:Strengthofparachutelines. Measurestrengthofaparachutelineat6positions Physicalreasonstobelievethatstrengthvarieslinearlywithposition Modelby y i = + 1 X i + i ,where X istheposition, y isthe strength,and i indexesthemeasurement Whatifhave6lines,eachwith6observations? Measurementsnestedinline suggests: Y ij = + 1 X ij + j + ij = ( + j )+ 1 X ij + ij , where j indexestheline. Interceptvariesbetweenlines,butslopedoesnot c 2011Dept.Statistics(IowaStateUniversity) Stat511section23 2/26 1 2 3 4 5 6 9 1 1 1 1 2 1 3 Position S t r e n g t h Parachute line strength c 2011Dept.Statistics(IowaStateUniversity) Stat511section23 3/26 Randomcoefficientregressionmodelsallowslopetoalsovary Y ij =( + j )+( 1 + j 1 ) X ij + ij u =[ 10 , 11 , 20 , 21 ,..., 60 , 61 ] u = 1100 ... 1200 ... 1300 ... 1400 ... 1500 ... 1600 ... 00110 ... .. . .. . .. . .. . .. . 0001 6 36 12 c 2011Dept.Statistics(IowaStateUniversity) Stat511section23 4/26 j j 1 N , G G = 2 01 01 2 1 sometimesseemodelwrittenas: Y ij = j + 1 j X ij + ij , j j 1 N 1 , G R usuallyassumed 2 e I . = ZGZ + R isquitecomplicated,canwriteoutbutnot enlightening.Features: Var Y ij notconstant,dependson X ij ,evenif R is 2 e I Cov Y ij , Y i j notconstant,dependson X ij and X i j Cov Y ij , Y i j = ,sinceobs.ondifferentlinesassumedindependent so isblockdiagonal,withnon-zeroblocksforobs.onsameline c 2011Dept.Statistics(IowaStateUniversity) Stat511section23 5/26 Customarytoincludeaparameterforthecovariancebetween interceptandsloperandomeffects. ifomit,thenmodelisnotinvarianttotranslationofX i.e.,fixedeffectpartoftheregressionisthesamemodelevenif shift X ,e.g. X 3 . randomeffectspartisthesameonlyifincludethecovariance someparametervalueswillchangeif X shifted,butstructurestays thesame. Canextendmodelinatleasttwoways: 1. Moreparametersinregressionmodel e.g.quadraticpolynomial: Y ij = j + 1 j X ij + 2 j X 2 ij + ij , Example:AllanTrappsMS.Longevityofstoredseed,quadratic, 2833seedlotsofmaize,eachwith3to7observations. 2. Morelevelsofrandomeffects 6Measurementsperline,4linesperparachute MeasurementsnestedwithinLines,LinesnestedwithinChutes c 2011Dept.Statistics(IowaStateUniversity) Stat511section23 6/26 RepeatedExperiments Insomescientificfields,itisexpectedthatyouwillrepeatthe entireexperiment...
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.

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23Models - Acollectionofpotentiallyusefulmodels

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